Modified Iterative Algorithms for Nonexpansive Mappings

Abstract

Let 𝐻 be a real Hilbert space, let 𝑆, 𝑇 be two nonexpansive mappings such that 𝐹(𝑆)∩𝐹(𝑇)≠∅, let 𝑓 be a contractive mapping, and let 𝐴 be a strongly positive linear bounded operator on 𝐻. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑆𝑦_𝑛, 𝑦_𝑛=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐴)𝑇𝑥_𝑛, 𝑛≥0 is a real number and {𝛼_𝑛}, {𝛽_𝑛} are two sequences in (0,1) satisfying the following control conditions: (C1) lim_𝑛→∞𝛼_𝑛=0, (C3) 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, then ‖𝑥_𝑛+1−𝑥_𝑛‖→0. We also discuss several special cases of this iterative algorithm.

1. Introduction

Let 𝐻 be a real Hilbert space. Recall that a mapping 𝑓∶𝐻→𝐻 is a contractive mapping on 𝐻 if there exists a constant 𝛼∈(0,1) such that

‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,𝑥,𝑦∈𝐻.(1.1) We denote by Π the collection of all contractive mappings on 𝐻, that is,

={𝑓∶𝐻⟶𝐻isacontractivemapping}.(1.2)

Let 𝑇∶𝐻→𝐻 be a nonexpansive mapping, namely,

‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,𝑥,𝑦∈𝐻.(1.3)

Iterative algorithms for nonexpansive mappings have recently been applied to solve convex minimization problems (see [1–4] and the references therein).

A typical problem is to minimize a quadratic function over the closed convex set of the fixed points of a nonexpansive mapping 𝑇 on a real Hilbert space 𝐻:

min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.4) where 𝐶 is a closed convex set of the fixed points a nonexpansive mapping 𝑇 on 𝐻, 𝑏 is a given point in 𝐻 and 𝐴 is a linear, symmetric and positive operator.

In [5] (see also [6]), the author proved that the sequence {𝑥_𝑛} defined by the iterative method below with the initial point 𝑥₀∈𝐻 chosen arbitrarily

𝑥_𝑛+1=1−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑏,𝑛≥0,(1.5) converges strongly to the unique solution of the minimization problem (1.4) provided the sequence {𝛼_𝑛} satisfies certain control conditions.

On the other hand, Moudafi [3] introduced the viscosity approximation method for nonexpansive mappings (see also [7] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contractive mapping on 𝐻. Starting with an arbitrary initial point 𝑥₀∈𝐻, define a sequence {𝑥_𝑛} in 𝐻 recursively by

𝑥_𝑛+1=1−𝛼_𝑛𝑇𝑥_𝑛+𝛼_𝑛𝑓𝑥_𝑛,𝑛≥0,(1.6) where {𝛼_𝑛} is a sequence in (0,1), which satisfies some suitable control conditions.

Recently, Marino and Xu [8] combined the iterative algorithm (1.5) with the viscosity approximation algorithm (1.6), considering the following general iterative algorithm:

𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑛≥0,(1.7) where 0<𝛾<𝛾/𝛼.

In this paper, we suggest a new iterative method for finding the pair of nonexpansive mappings. As an application and as special cases, we also obtain some new iterative algorithms which can be viewed as an improvement of the algorithm of Xu [7] and Marino and Xu [8]. Also we show that the convergence of the proposed algorithms can be proved under weaker conditions on the parameter {𝛼_𝑛}. In this respect, our results can be considered as an improvement of the many known results.

2. Preliminaries

In the sequel, we will make use of the following for our main results:

Lemma 2.1 (see [4]). Let {𝑠_𝑛} be a sequence of nonnegative numbers satisfying the condition 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛽_𝑛,𝑛≥0,(2.1) where {𝛼_𝑛}, {𝛽_𝑛} are sequences of real numbers such that (i){𝛼_𝑛}⊂[0,1] and ∑^∞𝑛=0𝛼_𝑛=∞,(ii)lim_𝑛→∞𝛽_𝑛≤0 or ∑^∞𝑛=0𝛼_𝑛𝛽_𝑛 is convergent.Then lim_𝑛→∞𝑠_𝑛=0.

Lemma 2.2 (see [9, 10]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.(2.2) Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.3 (see [2] (demiclosedness Principle)). Assume that 𝑇 is a nonexpansive self-mapping of a closed convex subset 𝐶 of a Hilbert space 𝐻. If 𝑇 has a fixed point, then 𝐼−𝑇 is demiclosed, that is, whenever {𝑥_𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑇)𝑥_𝑛} strongly converges to some 𝑦, it follows that (𝐼−𝑇)𝑥=𝑦, where 𝐼 is the identity operator of 𝐻.

Lemma 2.4 (see [8]). Let {𝑥_𝑡} be generated by the algorithm 𝑥_𝑡=𝑡𝛾𝑓(𝑥_𝑡)+(𝐼−𝑡𝐴)𝑇𝑥_𝑡. Then {𝑥_𝑡} converges strongly as 𝑡→0 to a fixed point 𝑥^∗ of 𝑇 which solves the variational inequality ⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥^∗−𝑥⟩≤0,𝑥∈𝐹(𝑇).(2.3)

Lemma 2.5 (see [8]). Assume 𝐴 is a strong positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

3. Main Results

Let 𝐻 be a real Hilbert space, let 𝐴 be a bounded linear operator on 𝐻, and let 𝑆, 𝑇 be two nonexpansive mappings on 𝐻 such that 𝐹(𝑆)∩𝐹(𝑇)≠∅. Throughout the rest of this paper, we always assume that 𝐴 is strongly positive.

Now, let 𝑓∈Π with the contraction coefficient 0<𝛼<1 and let 𝐴 be a strongly positive linear bounded operator with coefficient 𝛾>0 satisfying 0<𝛾<𝛾/𝛼. We consider the following modified iterative algorithm:

𝑥𝑦_𝑛+1𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑦_𝑛,=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0,(3.1) where 𝛾>0 is a real number and {𝛼_𝑛}, {𝛽_𝑛} are two sequences in (0,1).

First, we prove a useful result concerning iterative algorithm (3.1) as follows.

Lemma 3.1. Let {𝑥_𝑛} be a sequence in 𝐻 generated by the algorithm (3.1) with the sequences {𝛼_𝑛} and {𝛽_𝑛} satisfying the following control conditions: (C1)lim_𝑛→∞𝛼_𝑛=0,(C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.Then ‖𝑥_𝑛+1−𝑥_𝑛‖→0.

Proof. From the control condition (C1), without loss of generality, we may assume that 𝛼_𝑛≤‖𝐴‖⁻¹. First observe that ‖𝐼−𝛼_𝑛𝐴‖≤1−𝛼_𝑛𝛾 by Lemma 2.5.
Now we show that {𝑥_𝑛} is bounded. Indeed, for any 𝑝∈𝐹(𝑆)∩𝐹(𝑇), ‖‖𝑦_𝑛−𝑝‖‖=≤𝛼‖‖𝛼_𝑛𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑝‖‖≤𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑝)‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖+1−𝛼_𝑛𝛾‖‖𝑇𝑥_𝑛−𝑝‖‖=1𝛾−𝛼‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖+1−𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑝‖‖𝛾−𝛾𝛼𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖.(3.2) At the same time, ‖‖𝑥_𝑛+1−𝑝‖‖=≤𝛽‖‖𝛽_𝑛𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑆𝑦_𝑛−𝑝‖‖≤𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛‖‖𝑆𝑦_𝑛−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛‖‖𝑦_𝑛−𝑝‖‖.(3.3) It follows from (3.2) and (3.3) that ‖‖𝑥_𝑛+1−𝑝‖‖≤+𝛽_𝑛𝛼‖‖_𝑛𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛1−𝛾−𝛾𝛼𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖=1−1−𝛽_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖+𝛾−𝛾𝛼𝛼_𝑛1−𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖𝛾−𝛾𝛼𝛼_𝑛1−𝛽_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖𝛾−𝛾𝛼,(3.4) which implies that ‖‖𝑥_𝑛−𝑝‖‖≤max‖‖𝑥₀−𝑝‖‖,‖𝛾𝑓(𝑝)−𝐴𝑝‖𝛾−𝛾𝛼,𝑛≥0.(3.5) Hence {𝑥_𝑛} is bounded and so are {𝐴𝑇𝑥_𝑛} and {𝑓(𝑥_𝑛)}.
From (3.1), we observe that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖==‖‖‖‖𝛼𝛼_𝑛+1𝑛+1𝛾𝑓𝑥_𝑛+1+𝐼−𝛼_𝑛+1𝐴𝑇𝑥_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛‖‖+𝐼𝛾−𝛼𝑓_𝑛+1𝑥_𝑛+1−𝑓𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛≤𝛼_𝑛+1𝐴𝑇𝑥_𝑛+1−𝑇𝑥_𝑛+𝛼_𝑛−𝛼_𝑛+1𝐴𝑇𝑥_𝑛‖‖+||𝛼𝛾_𝑛+1‖‖𝑓𝑥_𝑛+1−𝑓𝑥_𝑛‖‖+1−𝛼_𝑛+1𝛾‖‖𝑇𝑥_𝑛+1−𝑇𝑥_𝑛‖‖≤𝛼_𝑛+1−𝛼_𝑛||‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝐴𝑇𝑥_𝑛‖‖+||𝛼𝛾_𝑛+1𝛼‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛼_𝑛+1𝛾‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=1−−𝛼_𝑛||‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝐴𝑇𝑥_𝑛‖‖𝛾−𝛾𝛼𝛼_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝐴𝑇𝑥_𝑛‖‖.(3.6) It follows that ‖‖𝑆𝑦≤_𝑛+1‖‖−𝑦𝑆𝑦_𝑛𝑛+1‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝛾−𝛾𝛼𝛼_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝐴𝑇𝑥_𝑛‖‖,(3.7) which implies, from (C1) and the boundedness of {𝑥_𝑛}, {𝑓(𝑥_𝑛)}, and {𝐴𝑇𝑥_𝑛}, that limsup_𝑛→∞‖‖𝑆𝑦_𝑛+1−𝑆𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.8) Hence, by Lemma 2.2, we have ‖‖𝑆𝑦_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.9) Consequently, it follows from (3.1) that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑆𝑦_𝑛−𝑥_𝑛‖‖=0.(3.10) This completes the proof.

Remark 3.2. The conclusion ‖𝑥_𝑛+1−𝑥_𝑛‖→0 is important to prove the strong convergence of the iterative algorithms which have been extensively studied by many authors, see, for example, [3, 6, 7].

If we take 𝑆=𝐼 in (3.1), we have the following iterative algorithm:

𝑥𝑦_𝑛+1𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0.(3.11) Now we state and prove the strong convergence of iterative scheme (3.11).

Theorem 3.3. Let {𝑥_𝑛} be a sequence in 𝐻 generated by the algorithm (3.11) with the sequences {𝛼_𝑛} and {𝛽_𝑛} satisfying the following control conditions: (C1)lim_𝑛→∞𝛼_𝑛=0,(C2)lim_𝑛→∞𝛼_𝑛=∞,(C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.Then {𝑥_𝑛} converges strongly to a fixed point 𝑥^∗ of 𝑇 which solves the variational inequality ⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥^∗−𝑥⟩≤0,𝑥∈𝐹(𝑇).(3.12)

Proof. From Lemma 3.1, we have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖⟶0.(3.13)
On the other hand, we have ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤=‖‖‖‖𝑥𝑥_𝑛+1𝑛+1−𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑇𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝛽𝑥_𝑛−𝑇𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛−𝑇𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖+1−𝛽_𝑛‖‖𝑦_𝑛−𝑇𝑥_𝑛‖‖+1−−𝛽_𝑛𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝐴𝑇𝑥_𝑛‖‖,(3.14) that is, ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤11−𝛽_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝐴𝑇𝑥_𝑛‖‖,(3.15) this together with (C1), (C3), and (3.13), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖=0.(3.16)
Next, we show that, for any 𝑥^∗∈𝐹(𝑇), limsup_𝑛→∞⟨𝑦_𝑛−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗⟩≤0.(3.17)
In fact, we take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗⟩=lim_𝑘→∞⟨𝑥_𝑛𝑘−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗⟩.(3.18) Since {𝑥_𝑛} is bounded, we may assume that 𝑥_𝑛𝑘0𝑥00086⇀𝑧, where “⇀” denotes the weak convergence. Note that 𝑧∈𝐹(𝑇) by virtue of Lemma 2.3 and (3.16). It follows from the variational inequality (2.3) in Lemma 2.4 that limsup_𝑛→∞⟨𝑥_𝑛−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗⟩=⟨𝑧−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗⟩≤0.(3.19) By Lemma 3.1 (noting 𝑆=𝐼), we have ‖‖𝑦_𝑛−𝑥_𝑛‖‖⟶0.(3.20) Hence, we get limsup_𝑛→∞⟨𝑦_𝑛−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗⟩≤0.(3.21)
Finally, we prove that {𝑥_𝑛} converges to the point 𝑥^∗. In fact, from (3.2) we have ‖‖𝑦_𝑛−𝑥^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖.(3.22) Therefore, from (3.16), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛽_𝑛𝑥_𝑛−𝑥^∗)+(1−𝛽_𝑛𝑦_𝑛−𝑥^∗)‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²=𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝛼_𝑛(𝛾𝑓𝑥_𝑛−𝐴𝑥^∗)+(𝐼−𝛼_𝑛𝐴)(𝑇𝑥_𝑛−𝑥^∗)‖‖²≤=𝛽1_𝑛‖‖−𝑥_𝑛2𝛼−_𝑛𝑥^∗‖‖²+1−𝛽_𝑛(1−𝛼_𝑛𝛾)²‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛⟨𝛾𝑓𝑥_𝑛−𝐴𝑥^∗,𝑦_𝑛−𝑥^∗⟩+2𝛼_𝑛𝛾+1−𝛽_𝑛𝛼^2𝑛𝛾²‖‖𝑥^𝑛−𝑥^∗‖‖²+2𝛼_𝑛⟨𝛾𝑓𝑥_𝑛−𝛾𝑓𝑥^∗,𝑦_𝑛−𝑥^∗⟩≤1−⟨2𝛾𝛼𝑓_𝑛𝑥^∗−𝐴𝑥^∗,𝑦_𝑛−𝑥^∗⟩𝛾+1−𝛽_𝑛𝛼^2𝑛𝛾²‖‖𝑥^𝑛−𝑥^∗‖‖²≤+12𝛼−_𝑛𝛾2𝛼𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝑦_𝑛−𝑥^∗‖‖+2𝛼_𝑛⟨𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑦_𝑛−𝑥^∗⟩𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝛼^2𝑛𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼^2𝑛𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖+2𝛼_𝑛⟨𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑦_𝑛−𝑥^∗⟩.(3.23) Since {𝑥_𝑛}, 𝑓(𝑥^∗) and 𝐴𝑥^∗ are all bounded, we can choose a constant 𝑀>0 such that 1𝛾−𝛾𝛼1−𝛽^𝑛𝛾²2‖‖𝑥^𝑛−𝑥^∗‖‖²+𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝛾𝑓𝑥^∗−𝐴𝑥^∗‖‖≤𝑀,𝑛≥0.(3.24) It follows from (3.23) that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−2𝛾−𝛼𝛾𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛾−𝛼𝛾𝛼_𝑛𝛿_𝑛,(3.25) where 𝛿_𝑛=𝛼_𝑛𝑀+1𝛾−𝛾𝛼⟨𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑦_𝑛−𝑥^∗⟩.(3.26) By (C1) and (3.17), we get limsup_𝑛→∞𝛽_𝑛≤0.(3.27) Now, applying Lemma 2.1 and (3.25), we conclude that 𝑥_𝑛→𝑥^∗. This completes the proof.

Taking 𝑇=𝐼 in (3.1), we have the following iterative algorithm:

𝑥𝑦_𝑛+1𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑦_𝑛,=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑥_𝑛,𝑛≥0.(3.28)

Now we state and prove the strong convergence of iterative scheme (3.28).

Theorem 3.4. Let {𝑥_𝑛} be a sequence in 𝐻 generated by the algorithm (3.28) with the sequences {𝛼_𝑛} and {𝛽_𝑛} satisfying the following control conditions: (C1)lim_𝑛→∞𝛼_𝑛=0,(C2)lim_𝑛→∞𝛼_𝑛=∞,(C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.Then {𝑥_𝑛} converges strongly to a fixed point 𝑥^∗ of 𝑆 which solves the variational inequality ⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥^∗−𝑥⟩≤0,𝑥∈𝐹(𝑆).(3.29)

Proof. From Lemma 3.1, we have ‖‖𝑥_𝑛−𝑆𝑦_𝑛‖‖⟶0.(3.30) Thus, we have ‖‖𝑥_𝑛−𝑆𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑆𝑦_𝑛‖‖+‖‖𝑆𝑦_𝑛−𝑆𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑆𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖−𝑆𝑦_𝑛‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛‖‖+‖‖𝐴𝑥_𝑛‖‖⟶0.(3.31) By the similar argument as (3.17), we also can prove that limsup_𝑛→∞⟨𝑦_𝑛−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗⟩≤0.(3.32) From (3.28), we obtain ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛽_𝑛𝑥_𝑛−𝑥^∗)+(1−𝛽_𝑛(𝑆𝑦_𝑛−𝑥^∗)‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝑆𝑦_𝑛−𝑥^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²=𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝛼_𝑛(𝛾𝑓𝑥_𝑛−𝐴𝑥^∗)+(𝐼−𝛼_𝑛𝐴)𝑥_𝑛−𝑥^∗)‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛1−𝛼_𝑛𝛾²‖‖𝑥_𝑛−𝑥^∗‖‖²+2⟨𝛾𝑓𝑥_𝑛−𝐴𝑥^∗,𝑦_𝑛−𝑥^∗⟩.(3.33) The remainder of proof follows from the similar argument of Theorem 3.3. This completes the proof.

From the above results, we have the following corollaries.

Corollary 3.5. Let {𝑥_𝑛} be a sequence in 𝐻 generated by the following algorithm 𝑥𝑦_𝑛+1𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0,(3.34) where the sequences {𝛼_𝑛} and {𝛽_𝑛} satisfy the following control conditions: (C1)lim_𝑛→∞𝛼_𝑛=0, (C2)lim_𝑛→∞𝛼_𝑛=∞, (C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.Then {𝑥_𝑛} converges strongly to a fixed point 𝑥^∗ of 𝑇 which solves the variational inequality ⟨(𝐼−𝑓)𝑥^∗,𝑥^∗−𝑥⟩≤0,𝑥∈𝐹(𝑇).(3.35)

Corollary 3.6. Let {𝑥_𝑛} be a sequence in 𝐻 generated by the following algorithm 𝑥𝑦_𝑛+1𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑦_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑥_𝑛,𝑛≥0,(3.36) where the sequences {𝛼_𝑛} and {𝛽_𝑛} satisfy the following control conditions: (C1)lim_𝑛→∞𝛼_𝑛=0, (C2)lim_𝑛→∞𝛼_𝑛=∞, (C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.Then {𝑥_𝑛} converges strongly to a fixed point 𝑥^∗ of 𝑆 which solves the variational inequality ⟨(𝐼−𝑓)𝑥^∗,𝑥^∗−𝑥⟩≤0,𝑥∈𝐹(𝑆).(3.37)

Remark 3.7. Theorems 3.3 and 3.4 provide the strong convergence results of the algorithms (3.11) and (3.28) by using the control conditions (C1) and (C2), which are weaker conditions than the previous known ones. In this respect, our results can be considered as an improvement of the many known results.